Modern cosmology has witnessed a remarkable success over the last several years and the credit goes a large amount of observational data coming from different astronomical surveys. With the increasing sensitivity in the astronomical data, undoubtedly, we have been able to put stringent constraints on the cosmological parameters, and at the same time we are facing new challenges. The \(H_0\)  and \(S_8\)  tensions have emerged as two biggest challenges for the standard \(\Lambda \) CDM cosmology demanding its revision, and the Planck preference of a closed universe model (see Refs. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}) further argues that the curvature of the universe can play a vital role in the estimations of the cosmological parameters. Even though the earlier investigations were in agreement with a spatially flat universe, however, there is no reason to stick to this point [7]} and the effects of curvature on the cosmological models and the key cosmological parameters should be investigated.
Thus, the main theme of this work is the curvature of the universe and its impacts on extended cosmologies with a special focus on the cosmological tensions.
